Solve for n n^2+100=19

$n^{2} + 100 = 19$

Step 1

Move all terms not containing $n$ to the right side of the equation.

Subtract $100$ from both sides of the equation.

$n^{2} = 19 - 100$

Subtract $100$ from $19$ .

$n^{2} = - 81$

Step 2

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$n = \pm \sqrt{- 81}$

Step 3

Simplify $\pm \sqrt{- 81}$ .

Rewrite $- 81$ as $- 1 (81)$ .

$n = \pm \sqrt{- 1 (81)}$

Rewrite $\sqrt{- 1 (81)}$ as $\sqrt{- 1} \cdot \sqrt{81}$ .

$n = \pm \sqrt{- 1} \cdot \sqrt{81}$

Rewrite $\sqrt{- 1}$ as $i$ .

$n = \pm i \cdot \sqrt{81}$

Rewrite $81$ as $9^{2}$ .

$n = \pm i \cdot \sqrt{9^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$n = \pm i \cdot 9$

Move $9$ to the left of $i$ .

$n = \pm 9 i$

Step 4

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$n = 9 i$

Next, use the negative value of the $\pm$ to find the second solution.

$n = - 9 i$

The complete solution is the result of both the positive and negative portions of the solution.

$n = 9 i, - 9 i$